Bell Work:
Given the sets L = {0, 1, 2, 3},
M = {5, 6, 7}, and N = {0, 1},
are the following statements
true or false?
(a) 6
L
(b) 0
N
Answer:
(a) False
(b) True
Lesson 61:
Subsets, Subsets of the
Set of Real Numbers
If all the members of one set
are also members of a
second set, the first set is
said to be a subset of the
second set.
If we have the two sets
B = {1, 2} A = {1, 2, 3}
Then we can say that set B is
a subset of set A because
all the members of set B are
also members of set A.
We use the symbol to
mean is a subset of.
Therefore, we can write
B
A
Which is read as “set B is a
subset of set A”
Since there are members of
set A which are not
members of set B, we say
“set B can be properly
contained in set A” or simply
“set B is a proper subset of
set A.”
The slash can be used to
negate the symbol . We
can write
A
B
Which is read “set A is not a
subset of set B” because all
the members of set A are
not members of set B
The set that has no members is
defined to be a proper subset of
every set that has members and
to be an improper subset of
itself. This set is called the empty
set or the null set and can be
designated by using either of the
symbols shown here.
{ } empty set  null set
Example:
Given the sets D = {0, 1, 2}.
E = {1, 2, 3, …}, and G = {1, 3, 5},
tell which of the following
assertions are true and which are
false and why.
(a) E
G (b) G
E (c) D
E
Answer:
(a) False, all members of set E
are not members of set G.
(b) True, All members of set G
are members of set E
(c) False, Zero, which is an
element of set D, is not a
member of set E.
Real Numbers
Rational Numbers
Real numbers that can be expressed
as the fraction of two integers: ½, 0.3,
-2/3 etc.
Irrational Numbers
Real numbers that
cannot be
expressed as the
fraction of two
integers, such as:
√5, √2. -√2, 2√2, π,
3π, etc.
Integers
Set of whole numbers and the
Opposites: …-1, 0, 1, …
Whole Numbers
Set of natural numbers & 0:
0, 1, 2, 3, 4, …
Natural or
Counting Numbers
1, 2, 3, 4, ….
Symbols to be aware of.
Useful facts about irrational
numbers that can be used in
problem sets.
1. The product of an irrational
number and a nonzero rational
number is an irrational number.
2. The sum of a rational number
and an irrational number is an
irrational number.
Example:
½
{What subsets of the
real numbers}?
Answer:
This asks that we identify the
subsets of the real numbers
of which the number ½ is a
member.
Rational and Real Numbers
Practice:
5
{What subsets of real
numbers}?
Answer:
Natural, Whole, Integer,
Rational, and Real
Practice:
3√2 {What subsets of real
numbers}?
Answer:
Irrational Numbers and Real
Numbers
Practice:
Tell whether the following
statements are true or false and
then explain why:
(a) {Reals}
{Integers}
(b) {Irrationals}
{Reals}
(c) {Irrationals}
{Rationals}
(d) {Wholes}
(e) {Integers}
{Naturals}
{Reals}
Answer:
(a) False
(b) True
(c) False
(d) False
(e) True
HW: Lesson 61 #1-30